stats distributions

Intuitively, you reject the null hypothesis if your observations give compelling evidence that your hypothesis was false. You work this out by computing the probability of your observation given the model provided by H_0 (p-value), if this is lower than a set threshhold (your significance level), you deem that observation very unlikely given H_0, you reject the null hypothesis.

You can think of the H_0 as an "initial guess" for a model, you decide whether an observation given would be typical from that model, and if not you judge that model as most likely being incorrect.

You fail to reject (not accept) the null hypothesis if there is no such compelling evidence and the p-value exceeds the significance level. You then judge the observation as being typical from the model. This does not mean the model is necessarily correct, you may have just "got lucky" with the observation being unsuspicious, hence why you are not accepting H_0 but rather failing to reject it. You could say the hypothesis testing is better for judging if models are incorrect rather than judging whether they are correct.